Select the correct answer.

[tex]$\angle A$[/tex] and [tex]$\angle B$[/tex] are complementary angles of right triangle [tex]$ABC$[/tex]. If [tex]$\cos A = 0.83$[/tex] and [tex]$\cos B = 0.55$[/tex], what is [tex]$\sin A + \sin B$[/tex]?

A. [tex]$0.28$[/tex]
B. [tex]$1$[/tex]
C. [tex]$0.38$[/tex]
D. [tex]$1.38$[/tex]



Answer :

To solve for [tex]\(\sin A + \sin B\)[/tex] given that [tex]\(\cos A = 0.83\)[/tex] and [tex]\(\cos B = 0.55\)[/tex], we can use the trigonometric identity:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
for each angle [tex]\(\theta\)[/tex].

First, we find [tex]\(\sin A\)[/tex]. Given [tex]\(\cos A = 0.83\)[/tex]:
[tex]\[ \sin^2(A) + \cos^2(A) = 1 \][/tex]
[tex]\[ \sin^2(A) + 0.83^2 = 1 \][/tex]
[tex]\[ \sin^2(A) + 0.6889 = 1 \][/tex]
[tex]\[ \sin^2(A) = 1 - 0.6889 \][/tex]
[tex]\[ \sin^2(A) = 0.3111 \][/tex]
[tex]\[ \sin(A) = \sqrt{0.3111} \approx 0.5578 \][/tex]

Next, we find [tex]\(\sin B\)[/tex]. Given [tex]\(\cos B = 0.55\)[/tex]:
[tex]\[ \sin^2(B) + \cos^2(B) = 1 \][/tex]
[tex]\[ \sin^2(B) + 0.55^2 = 1 \][/tex]
[tex]\[ \sin^2(B) + 0.3025 = 1 \][/tex]
[tex]\[ \sin^2(B) = 1 - 0.3025 \][/tex]
[tex]\[ \sin^2(B) = 0.6975 \][/tex]
[tex]\[ \sin(B) = \sqrt{0.6975} \approx 0.8352 \][/tex]

Finally, we add [tex]\(\sin A\)[/tex] and [tex]\(\sin B\)[/tex]:
[tex]\[ \sin A + \sin B \approx 0.5578 + 0.8352 = 1.392928045119877 \][/tex]

By observing the options given:
A. [tex]\(0.28\)[/tex]
B. [tex]\(1\)[/tex]
C. [tex]\(0.38\)[/tex]
D. [tex]\(1.38\)[/tex]

The closest option to our calculated sum of [tex]\(\sin A + \sin B\)[/tex] is:
[tex]\[ \boxed{1.38} \][/tex]